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A Financial Model of Foreign Exchange ExposureAuthor(s): Christine R. HekmanSource: Journal of International Business Studies, Vol. 16, No. 2 (Summer, 1985), pp. 83-99Published by: Palgrave Macmillan JournalsStable URL: http://www.jstor.org/stable/154656

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A FINANCIAL MODEL OF FOREIGN EXCHANGEEXPOSUREChristineR. Hekman*Duke University

Abstract. The paper describes a model of foreign exchange ex-posure. This is defined as the sensitivity of a specific investment'svalue in reference currency to changes in exchange rate forecasts.This sensitivity may result because some share of the investmentcash flows are denominated in foreign currency. Alternatively, ashare of cash flows denominated in reference currency which areaffected by future exchange rates can also generate sensitivity.The model integrates a general corporatevaluation frameworkwitha theory of expectations and a general model of corporate macro-economic relationships. Its contribution is in generalizing previousmodels of these relationships. It also provides a link betweenempirical work, theoretical descriptions of the exchange rate/relative price relationship, and corporate valuation theory.The model implies a fairly rich description of the corporate andeconomic characteristics which determine exposure. These de-scriptors may be used to explain differences in the responses toexchange rate changes of different companies, product lines, orindustries.

Foreign exchange risk is one of the unique complications of financialmanagement in an international environment. The values, in a firm's refer-ence currency, of many of its cash flows depend on foreign exchange ratesexpected to be in effect at settlement. Therefore, expectations aboutfuture exchange rates are important determinants of the expected futurevalues and thus the current values of such cash flows; further, variabilityin exchange rate forecasts may be a major source of variability in the cur-rent value. This variability is foreign exchange risk.* ChristineR. Hekman s AssociateProfessor n the FuquaSchool of Businessat DukeUniversity. Her area of researchand teaching specialty is internationalcorporatefi-nancial management.She has also servedas consultantto severalmajor corporationsandfinancial nstitutions.I am grateful for helpful comments by JamesHodder,Peter Jones, Donald Lessard,RichardLevich, David Mullins,an anonymous referee,and membersof the FinanceWorkshopat the Fuqua School of Business.Of course, any errorsremainmy own.Support for this work from the Associates of the HarvardBusinessSchool and theFuquaSchool of Business,DukeUniversity s gratefullyacknowledged.Date Received: September 30, 1981; Revised: September 6, 1983/June 5, 1984;Accepted: Sept-ember 28, 1984.

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JOURNAL OF INTERNATIONAL BUSINESS STUDIES, SUMMER 1985

The foreign exchange risk of any particularinvestment or project dependson both the variability of the exchange rate forecast and the specific sen-sitivity of the project's or investment's current value to changes in fore-casts. In turn, the investment-specific sensitivity is determined by suchfactors as the physical location of the investment, the structure of itsrevenues and costs, the industrial structure of the markets in which itoperates, and management's response to changing forecasts. This invest-ment-specific sensitivity to changes in exchange rate forecasts is foreignexchange exposure.This paper describes a model of foreign exchange exposure, that is, therelationship between changes in foreign exchange rate forecasts andchanges in the value, in reference currency, of cash flows whose values aresensitive to currency movements. While the relevant flows obviously in-clude those denominated in foreign currency, they also include futureoperating cash flows which are determined, among other things, by futureexchange rates. The contribution of the model is in its application to thelatter. Since the currency of settlement is irrelevant, future revenues andcosts may be nominally denominated in either the reference or foreigncurrencies; the model easily recognizes both. To simplify exposition, themodel here is applied only to changes in the values of investments whichare claims to be settled in foreign currency. However, the model may beused to describe the sensitivity to changes in exchange rate forecasts ofinvestments in riskless assets which are denominated in foreign currency,or simply to claims on reference currency flows whose amounts are in-fluenced by currency movements.Rather than being an ambitious attempt to describe the value of such in-vestments, the model serves to describe the effects on a given value ofchanges in exchange rate forecasts. Nor does the paper attempt an ex-tended discussion of the effects of foreign exchange risk on value. Sucha digression would require at least an implicit model of valuation whichmight specify the effects of exchange risk on the discount rate or, throughagency costs, on the expected values of the cash flows. Instead, the con-tribution here is a model of the sensitivity of value to changes in exchangerate forecasts, with an attendant description of specific determinants ofsensitivity.The implied description of the corporateand economic characteristics thatdetermine exchange rate sensitivity is rather detailed. These characteristicsmay, in turn, explain differences in the responses of competing companies,of product lines, or of industries to changes in exchange rates.The settlement of the claims for the investment under study will be in thecurrency of a single, small country.1 The value of these cash flows is mea-sured in nominal, reference currencyunits and is assumed to be determinedin perfect capital markets. Additional description of the capital market inwhich the investment is priced would be both extraneous to the model ofinvestment sensitivity and beyond the scope of the paper. Rather, thevalues of the claims are taken in the context of a given but unspecified

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MODEL OF FOREIGN EXCHANGE EXPOSURE

capital asset pricing model. Withvalue additivity, the aggregate nvestmentvalue is segregatedinto components of value within that same capital assetpricing framework.These components are the base case value of the claims if unlevered andunhedged, and the incremental values of outstanding foreign financing andforward foreign currency commitments. The model describes each com-ponent and shows that values depend on the exchange rate which initial-izes the stochastic generation of future rates. Comparativestatics demon-strate foreign exchange exposure as the sensitivity of the investment'svalue to changes in the initializing rate. The exposure or sensitivity of eachcomponent is described and assessed; aggregateexposure is the sum of thecomponent exposures.The research is by nature integrative. The model generalizes corporatevaluation theory in order to incorporate claims whose reference-currencyvalue is affected by changes in exchange rates; the relevance of expecta-tions is maintained. Classical macroeconomic relationships between ex-change rates and prices are imbedded in the model's structure although itallows deviation from strict classical macroeconomics. As a result, one ofthe most important characteristics of the model is its consistency with agenerally accepted macroeconomic structure.The model is also consistent with and may serve as a framework for morespecific models of foreign exchange exposure. These include Heckerman's[9] description of the relationship between the level of the real profitstream and the real terms of trade when sales and production quantitiesare constant; Shapiro's [17] extension of this paper by considering theeffects of changes in the terms of trade on nominal profits when quantitiesare variable; and Levi's [13] linkage of sales and financial effects in amulticurrency world based on modeling the effects of changes in the cur-rent spot exchange rate on the trading profits and portfolio returns ofexporting firms.Hodder [11] emphasized the domestic aspects of exposure by modelingthe effects of changes in the current spot exchange rate on the real rateof return to a 2-country firm and found that the proportion of net worthexposed is the regression coefficient of real returns on exchange ratechanges. He showed that this coefficient is determined by the probabilisticrelationships among nominal exchange rates and domestic prices, realexchange rates, and the domestic inflation rate, the levels of assets andnet worth, and the shares of net physical assets and initial net monetaryliabilities denominated in foreign currency. He specified the importantrelationship between nominal and real exchange rates generally as a ran-dom disturbance.The model described in this paper emphasizes the effects of changes inexpected exchange rates on foreign investment values. However, becausethe underlying specification recognizes the existence of competitors whomay be influenced by exchange rates, the model allows that purely do-mestic investments may be also affected by changes in exchange rate

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JOURNALOF INTERNATIONAL USINESSSTUDIES,SUMMER1985forecasts. A more comprehensive specification of production and demandcomponents enriches the more general approach of the earlier modelingexercises by identifying factors which determine company, product line,or industry-specific responses to exchange rate changes. Finally, the exten-sion from the single period treatment of the earlier models to a multiperiodframework allows for the specification of the effects of changes in expecta-tions or forecasts rather than simply the effects of changes in the currentspot rate. Although not pursued here, the extension provides a frameworkfor analyses of effects of changes in the term structure of expectationsand the effects of lags imbedded in the price-exchange rate relationshipover the term structure.

VALUATIONFRAMEWORKTo model exposure-that is, the sensitivity of an asset's value in referencecurrency to changes in exchange rate forecasts-we require an expressionfor the value of that asset. Here we consider the value of a single foreigncurrency investment, a claim to cash flows to be settled in the future inthat foreign currency.2 The foreign country is "small" in the sense thatthe currency exchange rate and foreign economic conditions affect neitherthe economy nor required rates of returns in the reference country.By assuming value additivity,3 we can disaggregate the value of the claiminto components. The central component is the base case value, the valueof the claims on future cash flows as if financed solely by equity and un-hedged. This value is adjusted to reflect the incremental values of out-standing debt and forward foreign currency commitments.4

VE = VCF + (1 -T)VD+VF (1)whereVE = the presentvalue,in the referencecurrency,of the flows

to equity holdersVCFu = the base casevalue;the presentvalue,in referencecurrency,of the firm's futureafter-taxoperatingcashflows grossofpaymentsto suppliersof capitalor to forwardcontractcommitmentsVD = the presentvalue,in referencecountrycurrency,ofthe flows to existing bond holders

T = the net effect of the corporate and personal tax structureon the gains to leverage 5This paper is organized around these components of value. It specifies amodel of each and reaggregates the components to describe the value ofthe asset. Without more specific assumptions about the underlying capitalasset pricing mechanism, the valuation equations represent only incom-plete descriptions of value. However, the absence of detailed specificationdoes not preclude an analysis of the implied exposure, to be measured asthe derivative of value with respect to changes in the vector of exchangerate forecasts. This exercise not only offers a model of exposure and itsdeterminants, but also demonstrates the exposure-reduction potential offoreign currency debt and forward contracts.

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MODEL OF FOREIGN EXCHANGE EXPOSURE

THEBASE CASEVALUEThe assumption that the claims will be settled in a single foreign currencyimplies that investor expectations of the equivalents of these flows in refer-ence currency depend on assessments both of the amount of the foreigncurrency flow and of the rate of exchange between the foreign and refer-ence currencies at the settlement date. These expected values of the refer-ence currency equivalents are discounted at a rate which reflects theirsystematic risk to obtain the reference currency value of the non-financialcash flows.6Investors discount the expected values of the base currency equivalencefrom the next period, t = 1, to the liquidation period, T. The implied rateof discount for the expected values of the base case cash flows reflects theunlevered and unhedged risk of a stream of reference currency flows. Byprior assumption, this rate reflects the systematic components of bothbusiness and foreign exchange risk-that variability in the net present valueof the cash flows resulting from changes in the expected values of ex-change rates.

T E(St Ct)VCFu = - (2)t=l (l+ro)t

where VCF = thecurrent ase-currencyalueof theafter-tax,non-financialash lowsSt = thevariablepotbase-currencyriceof theforeigncurrencyt timetCt = thevariablefter-taxoreign urrencyash lowsat timetro = the unleveredndunhedged ase-currencyostof equity

The settlement values of the foreign currency cash flows, Ct, may be de-scribed by a partial equilibrium model which assumes value-maximizingmanagerialbehavior and an industry setting of competitive supply and de-mand for the firm's production.7 With fixed production function para-meters and constant returns to scale, all variables within the model areshown to be functions of the foreign currency prices of traded goods andthe exchange rate. Assuming international price parity for traded goodsimplies that foreign currencycash flows can be fully described by exchangerates and traded goods prices in the reference currency. That is, manage-ment can describe the firm's current and expected future output prices,input prices, profits, and cash flows conditional on a set of foreign ex-change rate expectations.The relationship between foreign cash flows and prevailingexchange ratesderives from specific assumptions of production and demand functionstructures and input price elasticity. In particular,a Cobb-Douglas produc-tion function describes industry production. The model assumes that onefactor of production is in perfectly elastic supply and is freely traded in

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world markets. A second factor of production is inelastically supplied andis only traded locally.8 The output price, aggregate ncome, own price andincome elasticity, and cross,price, elasticities for substitute and comple-mentary products determine demand.Other industries may use both internationally mobile and immobile inputs.The use of internationally mobile inputs is a common link among firms inthe national economy when the relative prices of internationally mobilefactors are assumed to be fixed.9 Under this assumption, the price of eachtraded factor is a linear function of the foreign currency index of tradedgoods prices.It also holds that the foreign currency price of the internationally mobilefactor of production is a single, exogenous determinant of the cash flows.The sensitivity of the firm's cash flows to changes in this price is stronglyinfluenced by the shareof that good in input costs as well as the degree ofinelasticity with which the immobile factor of production is supplied.Finally, the price behavior of this factor links foreign exchange rate ex-pectations and foreign cash flow expectations, because the factor mobilityassumption implies effective international price arbitrage.To begin development of the model we rely both on the assumption ofmarket perfection, which implies dividend policy irrelevance, and theCobb-Douglas production assumption, which implies constant factor costshares. Together these imply that returns to capital are a constant propor-tion, a k, of total revenue.

Ct = akRt (3)where ak = theshare f totalcostswhich sreturnedo capital

Rt = the variableotalrevenuenperiodWith production, factor supply, and demand assumptions outlined above,total revenueis a function of the index of traded goods prices in the foreigncountry, P.

Rt = t (4)where 3 = a constant arameter

r7 = anaggregatelasticity f revenuewithrespecto changesin theforeign urrencyraded oodsprice ndexPt = the indexof traded oodprices t timet stated nthe

foreign urrencyThe parameter, 3, captures those determinants of total revenue which areinsensitive to changes in traded goods prices. These include the levels offirm and industry technology, the share of the traded good input in thetraded good price index, and the non-price determinants of demand forindustry output. The demand determinants include aggregate ncome andthe demand for substitutable and complementary products.The revenue elasticity, r7,relates proportionate changes in revenue to theindex of foreign traded goods prices. This sensitivity is a function of theresponsiveness of both quantity supplied and quantity demanded. The

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MODELOF FOREIGNEXCHANGEEXPOSURE

output response reflects the underlying factor proportions and factor sup-ply elasticities. The demand response reflects the effects of price changeson the joint price responses of substitutes and complements. The aggregaterevenue elasticity is a function of these subsidiaryelasticities.'?Under these assumptions, the index of traded goods prices is the only vari-able element of firm cash flow.

Ct = k Ptr (5)Constant parameters, ak and p, capture the remaining determinants ofcash flow through production and demand specifications which assumeconstant proportional relationships."The macroeconomic relationship among exchange rates, the foreign tradedgoods price index, and the reference currency index of traded goods pricesconstitutes the remaining link to reference currency equivalents. Thisspecification requiresthe prior assumption that the relative prices of worldtraded goods are constant and that internationally mobile factors of pro-duction are perfectly elastic. Together these imply perfect price parity fortraded goods as well as for indices of traded goods prices.12

St = Pt/Pt (6)where Pt = the index of tradedgoods prices n the referencecurrency ountry.The price parity relationship is an important transmissionand translationmechanism. With the additional assumption that the base country pricelevel is fixed, exchange rates completely determine variable foreign pricelevels. The restatement is substituted into equation 5 for a description ofthe foreign currency cashflow where the foreign exchange rate is the singlevariable determinant.

Ct = ?ak Pt St- (7)t = a^P p;77Slt^-~71 (7)Further substitution of equation 7 into equation 2 yields an expressionfor the base currency value of the foreign currency cash flow.

T E(ak PtrStl1- r)VCF = 2 (8t=l (1+ ro)tSince all parametersare constant, equation 8 can be rewritten to emphasizethe role of exchange rate expectations in the valuation process.

T ECSt1VCFu = Z ak?- Pt (9)t=l (l+ro)tThe important result is that the reference currency value of the base casecash flows depends on the expectations of future spot exchange rates. Ex-pectations on this single variable vector suffice to determine value becausethey directly imply expectations for both the price levels of traded goods

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JOURNAL OF INTERNATIONALBUSINESSSTUDIES,SUMMER1985in the foreign country and cash flows. In turn, the expectations are a func-tion of the stochastic process which generates exchange rate observations.In modeling this process, assume that exchange rates evolve as a randomwalk with drift u and that differences in the natural logarithms of succes-sive exchange rates are normally distributed.13

n St+ 1 - In St = M+ Et (10)whereat ' N(0,a2)

This assumption implies that exchange rate appreciations and deprecia-tions will proceed at a constant rate. In other words, the ratio of succes-sive exchange rates is lognormally distributed.14St+ls t- n-- (11)St

St+1s N(/, 02); -- A(i, a2)

StBy this assumption 3 factors determine exchange rates. The first is aninitial rate, So. The second is a drift term, ,, which reflects the anticipatedconstant differential between inflation in traded goods prices in the 2countries. The last factor is the intervening random term and the interven-ing random change.

St = So exp(s t) (12)whereSo = the initialexchangerate

The expectation of the variable exchange rate can now be rewritten as afunction of the initial exchange rate, So, time, and the t( 1 + ,/) -th mo-ment of the lognormal random variable.15E[St(1-7?)] = So(1-r) E[exp~ ? (1 -r7) t] (13)

With the substitution of equation 13 into equation 9 value is stated as afunction of constant parameters, the price index of traded goods in thereference currency country, expected exchange rate "returns" and theinitial exchange rate, So .16T akOPt7rE[exp~ * ( - 71) t So(1l )VCF = Z (14)t=l (1 +ro)t

In summary, fixed relationships among aggregatetraded goods price levels,exchange rate expectations and cash flows determine base case value inreference currency terms. Exchange rates are stochastic, and independentdrawingsfrom a distributionwith fixed parametersgenerate changes. Thus,exchange rate expectations and the base case value depend on the initializ-ing or conditioning exchange rate.

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MODELOF FOREIGNEXCHANGEEXPOSURETHEFOREIGNEXCHANGEEXPOSUREOF THE BASECASEVALUE

Foreign exchange exposure is the variation in reference currency valuewhich results from exchange rate changes. When the base country priceindex and the parametersof the stochastic exchange rate process are fixed,the only source of change is the initializing or conditioning exchangerate.17This change is the adjustment of the expected levels of exchange rates, anadjustment which affects all future periods. In effect, the straight butpotentially sloping yield curve is unilaterally raised or lowered by changesin the conditioning rate.The effect of changes in the initial exchange rate is the derivative of thevaluation equation with respect to changes in the conditioning rate.

dVCFU T ak/3Pt7E [exp' - t (1 -r)] S,(1-r)-1--_ = (1- r) z (15)

dSo t=l (1 +ro)tWith simple manipulation, exposure may be stated as either a coefficientor an elasticity.

dVCFU- = (1-r7) VCFU (16)dSo/SodVCF /VCFU = (1 - 7) (16')dSo So

The aggregate elasticity, ( - r), is the difference between unity and theelasticity of corporate revenue in foreign currency with respect to theprices of foreign traded goods. In turn, the revenue elasticity, r1, s a func-tion of production functions, input elasticities and demand elasticities.This formulation has the following intuitive content: the base currencyvalues of foreign cash flows are exposed to the extent that changes inforeign cash flows fail to offset exactly changes in exchange rate expecta-tions. For example, dollar values are exposed if foreign revenues cannotbe raised by the full extent of any foreign inflation/currency devaluation.Values are also exposed if such foreign revenue response should exceed de-valuation. However, in this case exposure is positive in that dollar valuesincrease with foreign currency devaluation. Conversely, foreign revenuesin a revaluing currency are exposed if the impact of foreign recession ex-ceeds the coincident appreciation of the foreign currency.Exposure as defined above is a sensitivity which is influenced by manyrelationships. These relationships can be captured as a single coefficientwhen the several important and restrictive assumptions of the model arevalid. One of these assumptions is that the response of revenue and cashflow to changes in foreign price conditions is independent of the timehorizon. Another is that shifts in exchange rate expectations are equivalent

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JOURNALOF INTERNATIONAL USINESSSTUDIES,SUMMER1985across the time horizon. With these assumptions the determinants of sen-sitivity are identical for all future time periods and exposure may be char-acterized by a simple coefficient.

THEADJUSTMENT ORTHEVALUE OF OUTSTANDINGDEBTOutstanding debt is a senior commitment which reduces the cash flow ac-cruing to equity at any given asset level. This income drain may be miti-gated by gains to leverageif the tax structure favors debt financing.lThe net incremental effect on the value of equity in each period is thevalue of debt service less the net value of tax shields. In determining thevalue of equity, the adjustment for the value of debt is deducted from thebase case value. All flows are evaluated in reference currency terms.

T [(1 -r)it + Dt] E(St)AVD= Z (17)t=1 (1+ rd)twhere AVD = the adjustment or the valueof debt

it = the tax deductible nterestpaymentat time tDt = the principalrepaymentat time trd = the pre-taxbasecurrencyreturncommensuratewiththe riskof the flows to bondholders

Equation (17) can be rewritten to recognize the earlier assumption thatexchange rates are a stochastic process.T [( - T)it + Dt] So E[exp( ?t)]AVD= Z (18)t=l (1 + rd)t

The effect of outstanding debt on foreign exchange exposure is simplythe derivative of the debt adjustment with respect to changes in So.dAVD T [(1 -T)it + Dt] E[exp( *t)]---. Z .. (19)dSo t=1 (+ rd)t

The elasticity form emphasizes the nature of foreign currency debt as acompletely exposed liability.dAVD/AVD = l (20)dSo S

Alternatively, debt can be viewed as a special case of the more generalvaluation equation 14 and exposure equations 15 through 16' where thecash flow elasticity, , is zero.19

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MODELOF FOREIGNEXCHANGEEXPOSURETHEADJUSTMENT OR THEVALUE OF OUTSTANDING

FORWARDFOREIGNEXCHANGECONTRACTSForward foreign exchange contracts serve as perfect hedges of value whencontract amounts equal the level of 'real' exposure and the assumed con-stancy of the parameters is maintained.20 Though this result has beenrecognized elsewhere, the development specifies the determinants of theincremental values of such contracts.Forward contracts are agreements to deliver a specified amount of foreigncurrency at a future time in exchange for a fixed amount of base countrycurrency. Outstanding forward foreign exchange commitments reduce orincrease the value of the firm as the values of the contracts are less than orgreater than zero. At the initiation of forward contracts the value of thefirm must be unaffected by fairly priced forward commitments. With thepassage of time, however, existing contracts may gain or lose value withchanges in the exchange rate expected to prevail at the correspondingfuture settlement date.The contractual forward rate is determined at the initiation of the contractand is assumed to be determined in a frictionless market. When appro-priately priced, the forward rate at time k for future period t is relatedto the expectation of the corresponding future spot rate by a risk pre-mium. This premium reflects the risk of forward contracting as assessed inthe base country capital markets.21

Fk,t = (+ rf)t Ek(St) (21)where Fk, t = the forwardrate at time k for foreignexchangeto be deliveredat time t.

Ek ( St) = the expectationat time k of the spot rateat time t.rf = the perperiodreturnassessedby basecountrycapitalmarkets or risk incurred n forwardcontracting.

The potential for arbitrage among securities in the base country's capitalmarket insures that the return to forward contracting is related to otherreference country capital market returns. In particular, the value of anyhedged cash flow must equal the combined values of an identical unhedgedcash flow and the hedge itself. At initiation of the contract, the hedge hasno value; the value of the hedged and unhedged flows must be identical.The relationship between a discount rate for hedged cash flows, the for-ward risk premium, and the rate for unhedged flows derives from thisequality.(1+rh) = (l+ru)(l+rf) (22)whereru = the simple,per periodrateof returnassessedby basecountry capitalmarketscommensuratewith the risk of

foreigncurrencycashflows to be receivedwith certaintybut unhedged.

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rh = the simple, perperiodrateof returnassessedby basecountrycapitalmarketscommensuratewith the riskof foreigncurrencycashflows to be receivedwithcertaintybut hedged.

With the passage of time forward contracts attain value since the claimson forward exchange are likely to be settled at rates different from thecontractual rates. Thus, existing forward contracts enhance or detractfrom total corporate value when gains or losses are expected at the settle-ment date.Thus, the values of outstanding claims on forward exchange reflect therequired rates of return as well as the expected future values of the claims.Specifically, the claims are agreements to deliver Gt units of foreign cur-rency at a future time t. The seller receives base-country currency at thecontractual exchange rate Ft in exchange. The value of outstanding for-ward contracts reflects the difference between the previously contractedrate of forward exchange and the rate at which the contract could bereversed. The latter is simply Ft, the current forward rate for deliveryat time t.22

T Gt (Ft - Ft)VF = - (23)t=1 ( + rh)twhere VF = the value,in referencecurrency,of outstanding orwardforeign exchangecontracts.

Ft = the contractual orwardrate on an existingcommitmentfor settlementat time t.Ft = the currentmarketratefor forwardexchangefordeliveryat time t,Gt = the contractamount,in foreigncurrencyunits;assumed o be madewith certainty

Like other elements of value, the forward contract is exposed. The degreeof exposure is the derivative of the forward contract value with respect tochanges in the initializing exchange rate.Following earlier derivation, which recognizes the stochastic process deter-mining spot and, by implication, forward exchange rates, the value of out-standing forward contracts may also be expressed as a function of theinitializing exchange rate.

T Fk, t -So exp(pf t)E[exp s ?t)]VH = Z Gt -- (24)t= 1 (1 + rh)tThe derivative of this value with respect to that exchange rate emphasizesthe role of exchange rate expectations in determining both the currentforward rate and the values of existing forward contracts.

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MODELOF FOREIGNEXCHANGEEXPOSUREdVF T Gt Ft/So- - Z (25)dSo t=l ( +rh)t

The formulation also can demonstrate the use of forward contracting as ahedging instrument. For example, consider the simple case of an unleveredcash flow matched by a forward contract commitment. Set the forwardcommitment at G* to match the level of foreign currency cash flow whichis exposed in each period. (This particularexample is distinguished by vari-ables and parameterswith asterisks(*).)G = (1- 77)E(C) (26)

Applying equation 15, the exposure of the base case value is a function ofthe 'real'exposure.dVCF* T E(C )-- = (1 -r) Z (27)

dSo t=l (l+r*)tThe exposure of the forward contract follows from equation 25.

dVF* T GIFt/So T E(CI)- = - 2 ---- =-(1-rI) -- (28)dSo t=l (1 + r)t t=l (1 + r*)t

In this case the changes in the base case value are exactly offset by changesin the value of the forward contract.dVE* dVCF* dVF*

= -- + -- = 0 (29)dSo dSo dSoFOREIGNEXCHANGEEXPOSURE

The equity ownership value is the sum of the individual elements of value.The primary element is the base case value, the value of the flows as if un-levered and unhedged. This value is adjusted to reflect the incrementalvalues of outstanding foreign financing and forward foreign currency com-mitments.VE = VFCu - AVD + VF (30)

Each component described above assumes perfect markets for goods andcapital and value maximizing managerialbehavior. Aggregationfrom equa-tions 30, 14, 18 and 24 yields a comprehensive description of equity value.T ?OkP7t7 E [exp ( -/)'s *? ] So( 1 )VE = - (31)t=l (l+ro)t

T [(1 - )it + Dt] E [exps ? t)] So

t=1(+r)

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T Fk, t-So ex t)expf- t)exp t)]+ 2 Gtt= 1 (1 + rh)t

Foreign exchange exposure is the derivative of equity value with respectto changes in exchange rate expectations. These changes are generated bychanges in the initializing exchange rate, So, when the parameters of thestochastic exchange rate process are assumed fixed.Since the initializing rate is the only variable in each element of value, thederivation is straightforward. The derivation also follows from equations30, 15, 19 and 25.

dVE T ak3Pt7E [exp(1 -). t] So- ) - 1- = (1-r7) ) - (32)dSo t=l (1 +ro)t

T [(1 - r)it + Dt] E [exp ( t)]-- -

t=l (1+rd)tT GtFf So1

_ C _--ft=1 (1 +rh)tWith some manipulation, the equation can be rewritten in elasticity terms.The revision highlights the aggregative nature of the model in that elasti-city of equity value with respect to exchange rates is a weighted sum ofthe components' elasticities.

T GtFtdVE/VE VCFu AVD t=1 (1 + rh)t- - = (1 -r7) - - (33)dSo So VE VE VE

The weights reflect the shares of each component in the total base cur-rency value of equity. Specifically, these include 1) the share of the basecase value in total equity value, 2) the share of the incremental adjustmentfor the value of debt in total equity value, and 3) the share of total equityvalue of the cash flows which have been hedged by forward commitment.(Note that the last component is not the value of existing forward con-tracts, but the value of the foreign currency under contract.)The individual elasticities, as described in each section above, are respec-tively (1 - 7), -1, and -1. In other words, the base case value may beless than fully exposed to the extent of local inflationary offset (1 > 7r> 0). Foreign currency debt and forward contracts are fully exposed.The formulation is appealing in its segregation of natural or operating ex-posure as represented by the exposure of the base case value. In essence,the elasticity (1 - ,) adjusts nominal cash flow to capture real exposure.

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questionwhich is rather nsignificantwhen the home countryis the UnitedStates.The pricingofvariabilitydepends,of course,on the structure f globalequilibrium.The underlying ssumptions f the modelare consistentwith the use of risk-adjusted iscountrates.We assume investors' constant relative risk aversionhere and show later that returnsare inde-pendentlydistributedovertime.7. Onlythe implications f theunderlying aluemaximizationmodel arepresentedhere.Importantassumptionsarerecognizedand the determinants f importantparameters re described.The de-tailed modelis describedn Hekman 10].8. For simplicity,the inputs are assumed o be eitherperfectlymobileor totally immobile.How-ever,nothing n the model's ogicor designprecludes xpansion o recognize ntermediatemobility.9. This is impliedby the "smallcountry"assumption ince the pricesof globallymobilefactorsaredeterminedexogenouslyand are unaffectedby events n the localeconomy.With hisassump-tion the theoretical rameworks consistentwith Dornbusch'smacroeconomicmodel [5] where:"The exchange rate now plays the key role of determiningrelativewages, relativeprices, andtherefore he allocationof resources."10. In the followingdevelopment andr are constantparameters. heyareindependentof boththe magnitudeof change n Pt and of time. Revenueresponses o changedpricesare complete neach period.These assumptionsarenot necessary or the model'sdevelopment,but simplifyboththe presentationand the result. In fact, time-dependentparameters re a complicationwhichwouldfacilitate he analysisof laggedresponses o changing conomicconditions.11. Constancyof these parameterss assumed or simplicityand in consistencywith the article'sfocus on the effects of exchangerate changes.Sincethe model'sconstructionnsures hat changesin the parameters re independent rom changes n Pt, parameter ariationscould be introducedandtreatedadditively.12. For simplicitywe also assume denticalpriceindex weightsin both countries.In the absenceof this assumptionrelativeprice levels would be related in constant proportionality o the ex-changerates.13. The assumptionof lognormality s useful but not required or furtherdevelopmentof thismodel. However, his specification s consistentwith the Shapiro 181 andAdlerandDumas [131reviewsof theoreticaland empiricalwork in this area. In particular, he empiricalanalysesde-scribed n Roll [16] and AdlerandLehmann 4] concludethat the deviations howno significantmeanreversion endencies or the greatmajorityof countrypairsandtimeperiodsexamined.14. The assumptionof lognormality s useful but not requiredfor furtherdevelopmentof thismodel.15. Momentsof the log normaldistributionare simple unctionsof the firstandsecondmomentsof the normaldistribution; ee Lindgren 14], page176. "Thekth moment. . .is expressible ntermsof the momentgeneratingunctionof X:

E(Xk) = exp (k,g + o2 k2/2)Withthispropertyequation13 reduces o

E[~t(1-7)] = So(1- ) exp {(1-r7)t/L + (1-r7)2t2a2/2} (13a)16. Note that the assumption f log normality educes he expectationof exchangerate"returns"to a function of the fixed parameters f the underlyingnormaldistribution. See equation[13a].)Subsequentdevelopment gnores hepassage f time andthe generationof exchangerates.It shouldbe clear that the assumedexchangerate processtogetherwith the reflection of errors, t, in re-vised cashflow expectations mplies ndependentandidenticallydistributed eturns o equity.17. Of coursethis change s only one of the severalwhichwe could consider. t is interestingbe-cause it implies the conditions underwhichgenerallyaccepteddefinitionsof exposureare valid.The effects of changes n other determinants f exchangerateexpectationsmightbe exploredinfutureresearch.18. See Miller 15] for discussion. ee alsoabove.Thedevelopmenthereis intentionallyunspecificsince tax treatmentof foreigndebt obligationsandthevaluationof suchtreatment s complexandperhapsndeterminant.19. See Adler and Dumas[2], for a moregeneralmodel of foreignbondvaluationwhichrelaxesthe assumption hat changes n the currentexchangerate shift expectationson all futureratesbyanequivalentamount.In this casethe cash flow elasticityof debtmaynot equalzero.20. It shouldbe carefullynoted that financial uturesprovide mperfecthedges n the more realis-tic caseof the fixed parameters eingallowed o change.

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MODELOF FOREIGNEXCHANGEEXPOSURE21. As above, capitalmarket tructurewithinwhichriskpremiaare determined s left unspecified.The issue of forwardcontractrisk premia s addressedn Hansenand Hodrick[7], Frankel[6],andStulz [19]. Theapproach akenbelowis to assumeconstancyof premia.22. Thepotentialfor forwardcontractdefaultmaybe incorporatedwithoutlossof generality.

REFERENCES1. Adler,M. and Dumas,B. "The Microeconomics f the Firm n anOpenEconomy."AmericanEconomicReview(February1977), 180-89.2. Adler,M. andDumas,B. "TheExposureof Long-TermForeignCurrencyBonds."JournalofFinance and Quantitative Analysis (November 1980), 973-994.3. Adler, M. and Dumas,B. "InternationalPortfolio Choice and CorporationFinance:A Syn-thesis." The Journal of Finance (June 1983), 925-984.4. Adler,M. and Lehmann,B. "Deviations rom Purchasing owerParity n the LongRun."TheJournal of Finance (December 1983), 1471-1487.5. Dornbusch,R. "Money,ExchangeRates,andEmployment."OpenEconomyMacroeconomics.New York:BasicBooks, 1980.6. Frankel,J. A. "The Diversifiabilityof ExchangeRisk."Journalof InternationalEconomics(August1979), 379-393.7. Hansen,L. P. and Hodrick,R. J. "ForwardExchangeRatesAs OptimalPredictorsof FutureSpot Rates: An EconometricAnalysis."Journalof PoliticalEconomy (October1980), 829-853.8. Haley, C. W. and Schall, L. D. The Theoryof FinancialDecisions.New York: McGraw-Hill,1973.9. Heckerman,D. "The ExchangeRisk of ForeignOperations." ournalof Business (January1972).

10. Hekman,C. R. "TheRealEffectsof ForeignExchangeRateChanges n a Competitive,Profit-Maximizing irm."Unpublishedmimeo,1983.11. Hodder,J. E. "Exposure o ExchangeRate Movements." ournalof InternationalEconomics(November1982), 375-86.12. Lessard,D. R. "Evaluating oreignProjects:An AdjustedPresentValueApproach." n D. R.Lessard, ed., International Financial Management, pp. 577-592. Boston: Warren,Gorham &Lamont,1979.13. Levi, M. "Economic Exposure."InternationalFinance, Chap. 13, pp. 323-57. New York:McGraw-Hill ookCompany,1983.14. Lindgren,B. W.StatisticalTheory.London:TheMacMillan ompany, econdedition, 1968.15. Miller,M. H. "DebtandTaxes."Journalof Finance(May1977), 261-275.16. Roll, R. "Violationsof PurchasingPower Parityand Their Implications or EfficientInter-nationalCommodityMarkets." n M. Sarnatand G. P. Szego, eds.,InternationalTradeandFinance,Vol. 1, Chap.6, pp. 133-176. Cambridge,MA:Ballinger ublishingCompany,1979.17. Shapiro,A. C. "ExchangeRate Changes,Inflation, and the Value of the MultinationalCor-poration." Journal of Finance (May 1975), 485-502.18. Shapiro,A. C. "WhatDoes Purchasing owerParityMean?" ournalof InternationalMoneyand Finance (1983), 2, 295-318.19. Stulz, R. M. "The ForwardExchangeRate and Macroeconomics."ournalof InternationalEconomics(May 1982), 285-299.

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